New York: Gordon and Breach Science Publishers. 800 p. $ 175.00 (1990).
From the preface: The solution of many problems relating to various areas of science and technology reduces to the calculation of integrals and the summation of series containing elementary and special functions. As is well known, this task is considerably simplified by means of appropriate handbook literature from which we should single out the world-famous “Bateman Manuscript Project” series “Higher transcendental functions. Vol. I, II, III.” New York: McGraw-Hill (1953; Zbl 0051.30303, Zbl 0052.29502, Zbl 0064.06302)] and “Tables of integral transforms, Vol. I, II.” New York: McGraw-Hill (1954; Zbl 0055.36401, Zbl 0058.34103)] by A. Erdélyi et al. and also the Tables of integrals, sums, series and products [English translation, corrected edition. New York: Academic Press (1980; Zbl 0521.33001)] by I. S. Gradshtein and I. M. Ryzhik.
Over several decades these handbooks have been reference manuals for theoretical and experimental physicists, research engineers, and specialists in applied mathematics and cybernetics. However, they contained only formulae up to the end of the 1940s: and this has led to the need for creating a more complete reference manual, in which new results are reflected. In this connection, the Russian originals of Volumes 1 and 2 of this series appeared in the early 1980s [Vol. I: Moscow: Nauka (1981; Zbl 0511.00044); Vol. II: Moscow: Nauka (1983; Zbl 0626.00033)]. These contain results in this area of mathematical analysis that have been published in recent years. This volume contains tables of indefinite and definite integrals, finite sums and series and includes the functions of Struve, Weber, Anger, Lommel, Kelvin, Airy, Legendre, Whittaker, the hypergeometric and elliptic functions, the Mathieu functions, the MacRobert function, the Meijer function, the Fox function and several others. The book also contains tables of representations of generalized hypergeometric functions, and tables of Mellin transforms of a wide class of elementary and special functions, combined with tables of special cases of the Meijer -function. Sections are included devoted to the properties of the hypergeometric functions, the Meijer -function and the Fox -function. The appendix contains supplementary material which can be used in the calculation of integrals and the summation of series.
The main text is preceded by a fairly detailed list of contents, from which the required formulae can be found. The notation used is, by and large, the generally accepted notation of the mathematical literature and is listed in the indices at the end of the book.
For the sake of compactness of exposition, abbreviated notation is used.
The Russian original has been reviewed (Moskva: Nauka 1986) in Zbl 0606.33001.